In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions.
It has many applications in sieve theory.
It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated a specific version of the conjecture in 1968.
[1] One version of the conjecture is as follows, and stating it requires some notation.
, the prime-counting function, denote the number of primes less than or equal to
denote the number of primes less than or equal to
Dirichlet's theorem on primes in arithmetic progressions then tells us that where
is Euler's totient function.
If we then define the error function where the max is taken over all
by Enrico Bombieri[2] and A. I. Vinogradov[3] (the Bombieri–Vinogradov theorem, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the generalized Riemann hypothesis.
[4] In 1986, Bombieri, Friedlander and Iwaniec generalized the Elliott-Halberstam conjecture, using Dirichlet convolution of arithmetic functions related to the von Mangoldt function.
A striking one is the result announced by Dan Goldston, János Pintz, and Cem Yıldırım,[6][7] which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16.
In November 2013, James Maynard showed that subject to the Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.
[8] In August 2014, Polymath group showed that subject to the generalized Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6.
[9] Without assuming any form of the conjecture, the lowest proven bound is 246.
The original Elliott-Halberstam conjecture is not clearly stated in their paper,[1] but can be inferred there from (1) page 59 and the comment above the Theorem on page 62.
denotes the logarithmic integral and